Why does bosonic string theory require 26 spacetime dimensions?

In addition to Chris Gerig's operator-language approach, let me also show how this magical number appears in the path integral approach.

Let $\Sigma$ be a compact surface (worldsheet) and $M$ a Riemannian manifold (spacetime). The string partition function looks like $$Z_{string}=\int_{g\in Met(\Sigma)}dg\int_{\sigma\in Map(\Sigma,M)}d\sigma\exp(iS(g,\sigma)).$$ Here $Met(\Sigma)$ is the space of Riemannian metrics on $\Sigma$ and $S(g,\sigma)$ is the standard $\sigma$-model action $S(g,\sigma)=\int_{\Sigma} dvol_\Sigma \langle d\sigma,d\sigma\rangle$. In particular, $S$ is quadratic in $\sigma$, so the second integral $Z_{matter}$ does not pose any difficulty and one can write it in terms of the determinant of the Laplace operator on $\Sigma$. Note that the determinant of the Laplace operator is a section of the determinant line bundle $L_{det}\rightarrow Met(\Sigma)$. The measure $dg$ is a 'section' of the bundle of top forms $L_g\rightarrow Met(\Sigma)$. Both line bundles carry natural connections.

However, the space $Met(\Sigma)$ is enormous: for example, it has a free action by the group of rescalings $Weyl(\Sigma)$ ($g\mapsto \phi g$ for $\phi\in Weyl(\Sigma)$ a positive function). It also carries an action of the diffeomorphism group. The quotient $\mathcal{M}$ of $Met(\Sigma)$ by the action of both groups is finite-dimensional, it is the moduli space of conformal (or complex) structures, so you would like to rewrite $Z_{string}$ as an integral over $\mathcal{M}$.

Everything in sight is diffeomorphism-invariant, so the only question is how does the integrand change under $Weyl(\Sigma)$. To descend the integral from $Met(\Sigma)$ to $Met(\Sigma)/Weyl(\Sigma)$ you need to trivialize the bundle $L_{det}\otimes L_g$ along the orbits of $Weyl(\Sigma)$. This is where the critical dimension comes in: the curvature of the natural connection on $L_{det}\otimes L_g$ (local anomaly) vanishes precisely when $d=26$. After that one also needs to check that the connection is actually flat along the orbits, so that you can indeed trivialize it.

Two references for this approach are D'Hoker's lectures on string theory in "Quantum Fields and Strings" and Freed's "Determinants, Torsion, and Strings".

I think this is standard in some String Theory textbooks:
The quantum operators form the Virasoro algebra, where the generators obey $[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}m(m^2-1)\delta_{m+n,0}$. Here "c" is the central charge, which is the space-time dimension we are working over. We need this algebra to interact appropriately with the physical states of the system (i.e. $L_m|\phi\rangle$ information), and only when $c=26$ do we guarantee that there are no negative-norm states in the complete physical system.

[Addendum] In the method I described, $c=26$ arises correctly as the critical dimension so that no absurdities occur. What I believe David Roberts is thinking about (in his comment below) is another way to get the same answer: You consider light-cone coordinates and write down the mass-shell condition (summing over the worldsheet dimension $D−2$), and you end up with the requirement $\frac{D-2}{24}=1$. In other words, $c=D=26$.

I am quite late answering this question, even though I followed it when it first appeared, but it must have slipped my mind. Anyway, it's been a while now and nobody seems to have mentioned my favourite (algebraic) reason for this.

In the covariant BRST quantisation of the bosonic string, the space of physical states can be interpreted as the relative semi-infinite cohomology group $H^\bullet(\mathfrak{V},\mathfrak{z};\mathfrak{M})$, where $\mathfrak{V}$ is the Virasoro algebra, $\mathfrak{z}$ is its centre and $\mathfrak{M}$ is a $\mathfrak{V}$-module in the category $\mathcal{O}_o$, the subcategory of category $\mathcal{O}$ consisting of graded modules with finite-dimensional homogeneous subspaces.

The standard complex computing semi-infinite cohomology is the tensor product $\mathfrak{M}\otimes\bigwedge^\bullet_{\frac\infty2}\mathfrak{V}'$ of $\mathfrak{M}$ with the semi-infinite forms on $\mathfrak{V}$. To compute relative cohomology we need to consider forms which are both horizontal and invariant relative to the centre. Now, it so happens that $\bigwedge^\bullet_{\frac\infty2}\mathfrak{V}'$ is a $\mathfrak{V}$-module where the central element acts with eigenvalue (central charge) $-26$, so that for the relative subcomplex to be nontrivial, the central charge of $\mathfrak{M}$ must be $+26$.

Now then why do people say that the bosonic string needs $26$ dimensions? This, which is actually imprecise, comes from the fact that when considering the conformal field theory of string propagating on $d$-dimensional Minkowski spacetime, the resulting (Fock) modules $\mathfrak{M}$ have central change $d$.

Why do I say that this is imprecise? Because the relation between the central charge and the dimension is very much dependent on the space on which the string is propagating. It is not inconceivable that there might exist (non-flat) spacetimes $M$ for which the Virasoro modules resulting from the conformal field theory of string propagating on $M$ (were this actually possible to compute) have a central charge which is not equal to the dimension of $M$.

Said differently, there certainly exist $\mathfrak{V}$-modules with central charge $26$ with no clear/known geometric interpretation at present, and there is no reason to discard their eventual interpretation in terms of geometries with dimension $\neq 26$.